9.4E: Sum-to-Product and Product-to-Sum Formulas (Exercises)

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For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

30. \(\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{4}\right)\)

31. \(2 \sin \left(\frac{2 \pi}{3}\right) \sin \left(\frac{5 \pi}{6}\right)\)

32. \(2 \cos \left(\frac{\pi}{5}\right) \cos \left(\frac{\pi}{3}\right)\)

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

33. \(\sin \left(\frac{\pi}{12}\right)-\sin \left(\frac{7 \pi}{12}\right)\)

34. \(\cos \left(\frac{5 \pi}{12}\right)+\cos \left(\frac{7 \pi}{12}\right)\)

For the following exercises, change the functions from a product to a sum or a sum to a product.

35. \(\sin (9 x) \cos (3 x)\)

36. \(\cos (7 x) \cos (12 x)\)

37. \(\sin (11 x)+\sin (2 x)\)

38. \(\cos (6 x)+\cos (5 x)\)